# 2015 Effective Beams

## Product description

The effective beam is the average of all scanning beams pointing at a certain direction within a given pixel of the sky map for a given scan strategy. It takes into account the coupling between azimuthal asymmetry of the beam and the uneven distribution of scanning angles across the sky. It captures the complete information about the difference between the true and observed image of the sky. The effective beams are, by definition, the objects whose convolution with the true sky produce the observed sky map.

Details of the beam processing are given in the respective pages for HFI and LFI.

The full algebra involving the effective beams for temperature and polarisation was presented in [1], and a discussion of its application to Planck data is given in the appropriate Planck-2013-IV[2], Planck-2015-A05[3] and Planck-2013-VII[4] papers. Relevant details of the processing steps are given in the Effective Beams section of this document.

### Comparison of the images of compact sources observed by Planck with FEBeCoP products

We show here a comparison of the FEBeCoP-derived effective beams, and associated point spread functions, PSF (the transpose of the beam matrix), to the actual images of a few compact sources observed by Planck, for all and frequency channels, as an example. We show below a few panels of source images organized as follows:

• Row #1- DX9 images of four objects with their galactic (l,b) coordinates shown under the color bar
• Row #2- linear scale FEBeCoP PSFs computed using input scanning beams, Grasp Beams, GB, for and B-Spline beams,BS, Mars12 apodized for the channels and the BS Mars12 for the sub-mm channels, for (see section Inputs below).
• Row #3- log scale of #2; PSF iso-contours shown in solid line, elliptical Gaussian fit iso-contours shown in broken line

### Histograms of the effective beam parameters

Here we present histograms of the three fit parameters - beam , ellipticity, and orientation with respect to the local meridian and of the beam solid angle. The sky is sampled (pretty sparsely) at 3072 directions which were chosen as HEALpix nside=16 pixel centers for and at 768 directions which were chosen as HEALpix nside=8 pixel centers for . These uniformly sample the sky.

Where beam solid angle is estimated according to the definition: $4 \pi \sum$(effbeam)/max(effbeam) i.e., $4 \pi \sum(B_{ij}) / max(B_{ij})$

Histograms for effective beam parameters
Histograms for effective beam parameters

### Sky variation of effective beams solid angle and ellipticity of the best-fit Gaussian

• The discontinuities at the Healpix domain edges in the maps are a visual artifact due to the interplay of the discretized effective beam and the Healpix pixel grid.

### Statistics of the effective beams computed using FEBeCoP

We tabulate the simple statistics of , ellipticity (e), orientation ($\psi$) and beam solid angle, ($\Omega$), for a sample of 3072 and 768 directions on the sky for and data respectively. Statistics shown in the Table are derived from the histograms shown above.

• The derived beam parameters are representative of the NSIDE 1024 and 2048 healpix maps (they include the pixel window function).
• The reported _eff are derived from the beam solid angles, under a Gaussian approximation. These are best used for flux determination while the the Gaussian fits to the effective beam maps are more suited for source identification.

Statistics of the FEBeCoP Effective Beams Computed with the BS Mars12 apodized for the channels and oversampled
frequency mean(fwhm) [arcmin] sd(fwhm) [arcmin] mean(e) sd(e) mean($\psi$) [degree] sd($\psi$) [degree] mean($\Omega$) [arcmin$^{2}$] sd($\Omega$) [arcmin$^{2}$] _eff [arcmin]
030 32.239 0.013 1.320 0.031 -0.304 55.349 1189.513 0.842 32.34
044 27.005 0.552 1.034 0.033 0.059 53.767 832.946 31.774 27.12
070 13.252 0.033 1.223 0.026 0.587 55.066 200.742 1.027 13.31
100 9.651 0.014 1.186 0.023 -0.024 55.400 105.778 0.311 9.66
143 7.248 0.015 1.036 0.009 0.383 54.130 59.954 0.246 7.27
217 4.990 0.025 1.177 0.030 0.836 54.999 28.447 0.271 5.01
353 4.818 0.024 1.147 0.028 0.655 54.745 26.714 0.250 4.86
545 4.682 0.044 1.161 0.036 0.544 54.876 26.535 0.339 4.84
857 4.325 0.055 1.393 0.076 0.876 54.779 24.244 0.193 4.63

#### Beam solid angles for the PCCS

• $\Omega_{eff}$ - is the mean beam solid angle of the effective beam, where beam solid angle is estimated according to the definition: $4 \pi \sum$(effbeam)/max(effbeam), i.e. as an integral over the full extent of the effective beam, i.e. $4 \pi \sum(B_{ij}) / max(B_{ij})$.
• from $\Omega_{eff}$ we estimate the $fwhm_{eff}$, under a Gaussian approximation - these are tabulated above
• $\Omega^{(1)}_{eff}$ is the beam solid angle estimated up to a radius equal to one $fwhm_{eff}$ and $\Omega^{(2)}_{eff}$ up to a radius equal to twice the $fwhm_{eff}$.
• These were estimated according to the procedure followed in the aperture photometry code for the PCCS: if the pixel centre does not lie within the given radius it is not included (so inclusive=0 in query disc).

 Band $\Omega_{eff}$[arcmin$^{2}$] spatial variation [arcmin$^{2}$] $\Omega^{(1)}_{eff}$ [arcmin$^{2}$] spatial variation-1 [arcmin$^{2}$] $\Omega^{(2)}_{eff}$ [arcmin$^{2}$] spatial variation-2 [arcmin$^{2}$] 30 1189.513 0.842 1116.494 2.274 1188.945 0.847 44 832.946 31.774 758.684 29.701 832.168 31.811 70 200.742 1.027 186.260 2.300 200.591 1.027 100 105.778 0.311 100.830 0.410 105.777 0.311 143 59.954 0.246 56.811 0.419 59.952 0.246 217 28.447 0.271 26.442 0.537 28.426 0.271 353 26.714 0.250 24.827 0.435 26.653 0.250 545 26.535 0.339 24.287 0.455 26.302 0.337 857 24.244 0.193 22.646 0.263 23.985 0.191

## Production process

FEBeCoP, or Fast Effective Beam Convolution in Pixel space[1], is an approach to representing and computing effective beams (including both intrinsic beam shapes and the effects of scanning) that comprises the following steps:

• identify the individual detectors' instantaneous optical response function (presently we use elliptical Gaussian fits of Planck beams from observations of planets; eventually, an arbitrary mathematical representation of the beam can be used on input)
• follow exactly the Planck scanning, and project the intrinsic beam on the sky at each actual sampling position
• project instantaneous beams onto the pixelized map over a small region (typically <2.5 diameter)
• add up all beams that cross the same pixel and its vicinity over the observing period of interest
• create a data object of all beams pointed at all N'_pix_' directions of pixels in the map at a resolution at which this precomputation was executed (dimension N'_pix_' x a few hundred)
• use the resulting beam object for very fast convolution of all sky signals with the effective optical response of the observing mission

Computation of the effective beams at each pixel for every detector is a challenging task for high resolution experiments. FEBeCoP is an efficient algorithm and implementation which enabled us to compute the pixel based effective beams using moderate computational resources. The algorithm used different mathematical and computational techniques to bring down the computation cost to a practical level, whereby several estimations of the effective beams were possible for all Planck detectors for different scan and beam models, as well as different lengths of datasets.

### Pixel Ordered Detector Angles (PODA)

The main challenge in computing the effective beams is to go through the trillion samples, which gets severely limited by I/O. In the first stage, for a given dataset, ordered lists of pointing angles for each pixel - the Pixel Ordered Detector Angles (PODA) are made. This is an one-time process for each dataset. We used computers with large memory and used tedious memory management bookkeeping to make this step efficient.

### effBeam

The effBeam part makes use of the precomputed PODA and unsynchronized reading from the disk to compute the beam. Here we tried to made sure that no repetition occurs in evaluating a trigonometric quantity.

One important reason for separating the two steps is that they use different schemes of parallel computing. The PODA part requires parallelisation over time-order-data samples, while the effBeam part requires distribution of pixels among different computers.

### Computational Cost

The computation of the effective beams has been performed at the NERSC Supercomputing Center. The table below shows the computation cost for FEBeCoP processing of the nominal mission.

 Channel 30 44 70 100 143 217 353 545 857 PODA/Detector Computation time (CPU hrs) 85 100 250 500 500 500 500 500 500 PODA/Detector Computation time (wall clock hrs) 7 10 20 20 20 20 20 20 20 Beam/Channel Computation time (CPU hrs) 900 2000 2300 2800 3800 3200 3000 900 1100 Beam/Channel Computation time (wall clock hrs) 0.5 0.8 1 1.5 2 1.2 1 0.5 0.5 Convolution Computation time (CPU hr) 1 1.2 1.3 3.6 4.8 4 4.1 4.1 3.7 Convolution Computation time (wall clock sec) 1 1 1 4 4 4 4 4 4 Effective Beam Size (GB) 173 123 28 187 182 146 132 139 124

The computation cost, especially for PODA and Convolution, is heavily limited by the I/O capacity of the disc and so it depends on the overall usage of the cluster done by other users.

## Inputs

In order to fix the convention of presentation of the scanning and effective beams, we show the classic view of the Planck focal plane as seen by the incoming photon. The scan direction is marked, and the toward the center of the focal plane is at the 85 deg angle w.r.t spin axis pointing upward in the picture.

Planck Focal Plane

### The Focal Plane DataBase (FPDB)

The FPDB contains information on each detector, e.g., the orientation of the polarisation axis, different weight factors, (see the instrument RIMOs):


### The scanning strategy

The scanning strategy, the three pointing angle for each detector for each sample: Detector pointings for the nominal mission covers about 15 months of observation from Operational Day () 91 to 563 covering 3 surveys and half.

### The scanbeam

The scanbeam modeled for each detector through the observation of planets. Which was assumed to be constant over the whole mission, though FEBeCoP could be used for a few sets of scanbeams too.

• : GRASP scanning beam - the scanning beams used are based on Radio Frequency Tuned Model (RFTM) smeared to simulate the in-flight optical response.
• : B-Spline, BS based on 2 observations of Mars.

(see the instrument RIMOs).

N times geometric mean of of all detectors in a channel, where N

 channel Cutoff Radii in units of fwhm fwhm of full beam extent 30 - 44 - 70 2.5 100 2.25 23.703699 143 3 21.057402 217-353 4 18.782754 sub-mm 4 18.327635(545GHz) ; 17.093706(857GHz)

### Map resolution for the derived beam data object

• $N_{side} = 1024$ for frequency channels
• $N_{side} = 2048$ for frequency channels

## Related products

### Monte Carlo simulations

FEBeCoP software enables fast, full-sky convolutions of the sky signals with the Effective beams in pixel domain. Hence, a large number of Monte Carlo simulations of the sky signal maps map convolved with realistically rendered, spatially varying, asymmetric Planck beams can be easily generated. We performed the following steps:

• generate the effective beams with FEBeCoP for all frequencies for dDX9 data and Nominal Mission
• generate 100 realizations of maps from a fiducial power spectrum
• convolve each one of these maps with the effective beams using FEBeCoP
• estimate the average of the Power Spectrum of each convolved realization, and 1 $\sigma$ errors

As FEBeCoP enables fast convolutions of the input signal sky with the effective beam, thousands of simulations are generated. These Monte Carlo simulations of the signal (might it be or a foreground (e.g. dust)) sky along with LevelS+Madam noise simulations were used widely for the analysis of Planck data. A suite of simulations were rendered during the mission tagged as Full Focalplane simulations, FFP#. For example FFP6

### Beam Window Functions

The Transfer Function or the Beam Window Function $W_l$ relates the true angular power spectra $C_l$ with the observed angular power spectra $\widetilde{C}_l$:

$W_l= \widetilde{C}_l / C_l \label{eqn:wl1}$

Note that, the window function can contain a pixel window function (depending on the definition) and it is {\em not the angular power spectra of the scanbeams}, though, in principle, one may be able to connect them though fairly complicated algebra.

The window functions are estimated by performing Monte-Carlo simulations. We generate several random realisations of the sky starting from a given fiducial $C_l$, convolve the maps with the pre-computed effective beams, compute the convolved power spectra $C^\text{conv}_l$, divide by the power spectra of the unconvolved map $C^\text{in}_l$ and average over their ratio. Thus, the estimated window function

$W^{est}_l = \lt C^{conv}_l / C^{in}_l \gt \label{eqn:wl2}$

For subtle reasons, we perform a more rigorous estimation of the window function by comparing C^{conv}_l with convolved power spectra of the input maps convolved with a symmetric Gaussian beam of comparable (but need not be exact) size and then scaling the estimated window function accordingly.

Beam window functions are provided in the RIMO.

#### Beam Window functions, Wl, for Planck mission

Beam Window functions, Wl, for channels
Beam Window functions, Wl, for channels

## File Names

The effective beams are provided by the as files containg maps of the beams. For the file names the following convention is used:

• Single beam query: beams_FFF_PixelNumber.fits
• FFF is the channel frequency (one of 30, 44, 70, 100, 143, 217, 353, 545, 857);
• PixelNumber is the number of the pixel to which the beam corresponds ($0$ - $12 \times N_{\rm side}^2 - 1$). For the $N_{\rm side}=1024$, for the $N_{\rm side}=2048$;
• Multiple beam query: beams_FFF_FirstPixelNumber-LastPixelNumber.zip
The compressed files contains a set of files with the beams for the pixels covereing the selected region. FFF as for sinle beam query. The naming convention for the beam files contained in the .zip file is the same as for single beam queries.
• FirstPixelNumber is the lowest pixel number for the area covered by the request;
• LastPixelNumber is the highest pixel number for the area covered by the request;

## File format

The files provided by the contain maps of the beams.

## References

1. Fast Pixel Space Convolution for Cosmic Microwave Background Surveys with Asymmetric Beams and Complex Scan Strategies: FEBeCoP, S. Mitra, G. Rocha, K. M. Górski, K. M. Huffenberger, H. K. Eriksen, M. A. J. Ashdown, C. R. Lawrence, ApJS, 193, 5-+, (2011).
2. Planck 2013 results. IV. Low Frequency Instrument beams and window functions, Planck Collaboration, 2014, A&A, 571, A4
3. Planck 2015 results. IV. LFI beams and window functions, Planck Collaboration, 2016, A&A, 594, A4.
4. Planck 2013 results. VII. HFI time response and beams, Planck Collaboration, 2014, A&A, 571, A7